##
**Numerical methods based on sinc and analytic functions.**
*(English)*
Zbl 0803.65141

Springer Series in Computational Mathematics. 20. New York, NY: Springer- Verlag. xv, 565 p. (1993).

This excellent monograph offers a self-contained presentation of the sinc method and its application to the numerical solution of integral and differential equations. This book will be the standard reference for the sinc method. It is of interest for mathematicians, computational scientists and graduate students.

Let \(h>0\) and \(\text{sinc} (x) : = (\pi x)^{-1} \sin (\pi x)\). Using the basis functions \[ S(k,h) (x) : = \text{sinc} \bigl( (x - kh)/h \bigr), \] a given function \(f\) bounded on the real line is approximated by the cardinal function \[ C(f,h) (x) : = \sum^ \infty_{k = - \infty} f(kh) S(k,h) (x). \] First, the approximation of \(f\) by means of \(C(f,h)\) was studied by de la Vallée Poussin and Whittaker. Later, Shannon’s sampling theorem gave an essential impulse to application of this theory in signal processing. The author has special merits in this topic, since he has studied the sinc method over 30 years intensively. Thus, many results presented in this book are new. Note that the sinc method is closely related to the approximation by translates, wavelet theory, and multiscale technique.

Basic facts on analytic functions, polynomial approximation, and Fourier technique are presented in the first two chapters. Chapter 3 deals with the approximation of \(f\) by \(C(f,h)\), where \(f\) is analytic on a strip containing the real line. Interpolation, quadrature, Fourier and Hilbert transforms, derivatives, and indefinite integrals are determined approximately. All of these procedures converge at exponential and close to optimal rate. Using a conformal mapping, the results of Chapter 3 are extended in Chapter 4 to approximations over a contour such that a finite or semi-infinite interval is a special case.

In Chapter 5, procedures related to sinc methods are discussed. Chapter 6 illustrates the application of sinc methods to the approximate solution of integral equations. The author considers nonlinear Volterra integral equations, Cauchy singular integral equations, convolution equations, Wiener-Hopf integral equations, and the inversion of Laplace transform. If there exists an analytic solution, then it is shown that an exponential convergence rate is reachable by sinc methods.

Finally, Chapter 7 demonstrates the use of sinc methods to obtain approximate solutions of ordinary and partial differential equations for both initial and boundary value problems. It is pointed out that Galerkin, finite element, spectral, and collocation methods are essential the same for the sinc methods, since they all yield nearly the same system of linear equations, whose solutions have the same order of accuracy.

Each section ends with some problems. Each chapter closes with historical remarks. This book is completed by a detailed list of references containing 296 items.

Let \(h>0\) and \(\text{sinc} (x) : = (\pi x)^{-1} \sin (\pi x)\). Using the basis functions \[ S(k,h) (x) : = \text{sinc} \bigl( (x - kh)/h \bigr), \] a given function \(f\) bounded on the real line is approximated by the cardinal function \[ C(f,h) (x) : = \sum^ \infty_{k = - \infty} f(kh) S(k,h) (x). \] First, the approximation of \(f\) by means of \(C(f,h)\) was studied by de la Vallée Poussin and Whittaker. Later, Shannon’s sampling theorem gave an essential impulse to application of this theory in signal processing. The author has special merits in this topic, since he has studied the sinc method over 30 years intensively. Thus, many results presented in this book are new. Note that the sinc method is closely related to the approximation by translates, wavelet theory, and multiscale technique.

Basic facts on analytic functions, polynomial approximation, and Fourier technique are presented in the first two chapters. Chapter 3 deals with the approximation of \(f\) by \(C(f,h)\), where \(f\) is analytic on a strip containing the real line. Interpolation, quadrature, Fourier and Hilbert transforms, derivatives, and indefinite integrals are determined approximately. All of these procedures converge at exponential and close to optimal rate. Using a conformal mapping, the results of Chapter 3 are extended in Chapter 4 to approximations over a contour such that a finite or semi-infinite interval is a special case.

In Chapter 5, procedures related to sinc methods are discussed. Chapter 6 illustrates the application of sinc methods to the approximate solution of integral equations. The author considers nonlinear Volterra integral equations, Cauchy singular integral equations, convolution equations, Wiener-Hopf integral equations, and the inversion of Laplace transform. If there exists an analytic solution, then it is shown that an exponential convergence rate is reachable by sinc methods.

Finally, Chapter 7 demonstrates the use of sinc methods to obtain approximate solutions of ordinary and partial differential equations for both initial and boundary value problems. It is pointed out that Galerkin, finite element, spectral, and collocation methods are essential the same for the sinc methods, since they all yield nearly the same system of linear equations, whose solutions have the same order of accuracy.

Each section ends with some problems. Each chapter closes with historical remarks. This book is completed by a detailed list of references containing 296 items.

Reviewer: M.Tasche (Rostock)

### MSC:

65T40 | Numerical methods for trigonometric approximation and interpolation |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65R20 | Numerical methods for integral equations |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

65Dxx | Numerical approximation and computational geometry (primarily algorithms) |

44A10 | Laplace transform |

45Exx | Singular integral equations |

45G10 | Other nonlinear integral equations |

### Keywords:

interpolation; Galerkin method; finite element method; spectral method; Cauchy singular integral equations; convolution equations; monograph; sinc method; cardinal function; Shannon’s sampling theorem; signal processing; translates; wavelet theory; multiscale technique; quadrature; Fourier and Hilbert transforms; derivatives; indefinite integrals; nonlinear Volterra integral equations; Wiener-Hopf integral equations; inversion of Laplace transform; exponential convergence rate; collocation method
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\textit{F. Stenger}, Numerical methods based on sinc and analytic functions. New York, NY: Springer-Verlag (1993; Zbl 0803.65141)

### Digital Library of Mathematical Functions:

§3.3(vi) Other Interpolation Methods ‣ §3.3 Interpolation ‣ Areas ‣ Chapter 3 Numerical MethodsEight-Point Formula ‣ §3.4(i) Equally-Spaced Nodes ‣ §3.4 Differentiation ‣ Areas ‣ Chapter 3 Numerical Methods

§3.5(i) Trapezoidal Rules ‣ §3.5 Quadrature ‣ Areas ‣ Chapter 3 Numerical Methods